Proposition 3

Let $C$ be a commutative ring, $E, F$ two finitely generated projective $C$-modules* and $u : E \to F$ and $v : F \to E$ two linear mappings; then*

The two mappings $(u, v) \mapsto \mathrm{Tr}(u \circ v), (u, v) \mapsto \mathrm{Tr}(v \circ u)$ of

into $C$ are $C$*-bilinear*; it therefore suffices to verify (20) when $u$ is of the form $x \mapsto \langle x, a^* \rangle b$ and $v$ of the form $y \mapsto \langle y, b^* \rangle a$, with $a \in E, a^* \in E^*, b \in F, b^* \in F^*$. But then $v \circ u$ is the mapping $x \mapsto \langle x, a^* \rangle \langle b, b^* \rangle a$ and $u \circ v$ the mapping $y \mapsto \langle y, b^* \rangle \langle a, a^* \rangle b$. Formula (17) shows that the values of the two sides of (20) are equal to $\langle a, a^* \rangle \langle b, b^* \rangle$.